> > The sum of two real numbers of the unit interval need not be a real> > number of the unit interval.>> Then the set of reals in the unit interval do not form a commutative> group under addition and thus cannot be a linear space, and thus cannot> be either the domain or codomain of any linear mapping.

It can and it is.>> > Nevertheless we have the same structure for reals, their> > representation as binary strings, and paths of the Binary Tree.>> WM claimed a linear mapping between the set of binomial sequences and> the set of paths of a Complete Infinite Binary Tree.>> Thus requires, among other things, that both sets have the structure of> linear spaces,

You are in error.Every sum and every product that is possible in the reals of the unitinterval is possible in the Binary Tree and vice versa. And that isall that is required.